To find the linear equation of a line passing through two points, we need to determine the slope (m) and the y-intercept (b). The general form of a linear equation is:

$y = mx + b$

Step 1: Find the slope (m)

The slope $m$ is calculated using the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Where:

• $(x_1, y_1) = (-1, 1)$
• $(x_2, y_2) = (1, 5)$

Now, substitute the points into the formula:

$m = \frac{5 - 1}{1 - (-1)} = \frac{4}{2} = 2$

So, the slope $m = 2$.

Step 2: Find the y-intercept (b)

We now use one of the points and the slope to find the y-intercept $b$. Using the point-slope form $y = mx + b$, substitute the slope and the coordinates of one of the points, say $(-1, 1)$.

$1 = 2(-1) + b$

Solve for $b$:

$1 = -2 + b$ $b = 3$

Step 3: Write the linear equation

Now that we know the slope $m = 2$ and the y-intercept $b = 3$, the linear equation of the line is:

$y = 2x + 3$

This is the required equation of the line.

Would you like to see a graphical representation of this line? Let me know if you'd like more details or have any questions.

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Here are 5 related questions for further practice:

1. What is the equation of a line passing through the points $(2, 4)$ and $(6, 12)$?
2. How do you determine the slope of a horizontal line?
3. If a line has a slope of 0, what can you say about its graph?
4. Find the equation of a line with a slope of 3 and passing through the point $(0, 2)$.
5. How do you find the equation of a line parallel to $y = 2x + 3$ but passing through $(3, 1)$?

Tip: When calculating slope, always be careful with the signs of the coordinates!